SUBJECTIVE PROBABILITIES ON
SUBJECTIVELY UNAMBIGUOUS EVENTS
Larry G. Epstein
Jiankang Zhang
February, 2000
Abstract
This paper suggests a behavioral de…nition of (subjective) ambiguity in an abstract
setting where objects of choice are Savage-style acts. Then axioms are described that
deliver probabilistic sophistication of preference on the set of unambiguous acts. In
particular, both the domain and the values of the decision-maker’s probability measure
are derived from preference. It is argued that the noted result also provides a decisiontheoretic foundation for the Knightian distinction between risk and ambiguity.
1. INTRODUCTION
1.1. Objectives
At least since Frank Knight, economists have often referred to a distinction between risk and
ambiguity.1 Roughly, risk refers to situations where the likelihoods of relevant events may
be represented by a probability measure, while ambiguity refers to situations where there
is insu¢ cient information available for the decision-maker to assign probabilities to events.
Ellsberg demonstrated that such a distinction is empirically meaningful. In particular, his
Paradox showed that behavior re‡ecting aversion to ambiguity is inconsistent with Savage’s
Epstein is at the University of Rochester, Rochester, N.Y. 14627, lepn@troi.cc.rochester.edu; Zhang
is at the Department of Economics, Carleton University, Ottawa, Ontario, Canada, K1S 5B6, Jiankang_
Zhang@Carleton.ca. This paper was written in part while Epstein was a¢ liated with U. Toronto and visiting
HKUST and while Zhang was a¢ liated with U. Western Ontario. We are indebted to the Canadian SSHRC,
to the NSF (grant SES-9972442) and to Carleton University for …nancial support and to Michelle Cohen,
Werner Ploberger, David Schmeidler, Peter Wakker and especially Mark Machina, Massimo Marinacci and
Uzi Segal for valuable discussions. Two (very patient) referees provided extensive and thoughtful comments
that led to major improvements. We are grateful to them and to Drew Fudenberg. A version of this paper
was …rst circulated in 1997.
1
Knight and many other authors use the term uncertainty rather than ambiguity. We follow Ellsberg and
use uncertainty as a comprehensive term. Thus in our terminology, every event/act/prospect is uncertain
and each is either risky or ambiguous (but not both).
subjective expected utility theory. In response, some generalizations of the Savage model
have been developed (most notably, by Schmeidler (1989) and Gilboa and Schmeidler (1989))
in order to resolve the Ellsberg Paradox and, more generally, to accommodate nonindi¤erence
to ambiguity.
All of this has proceeded without a formal de…nition of ambiguity for a general abstract
setting of the sort considered by Savage. The main objective of this paper is to provide such
a de…nition.
In order to motivate the paper further and to cast some light on the desiderata for a
de…nition of ambiguity, we describe two classes of applications that are developed below.
1.2. A Fully Subjective Theory of Probability
Savage’s expected utility theory is typically referred to as providing a subjective theory of
probability. That is because the probability measure underlies choice behavior. More precisely, it is derived from axioms on the preference ordering of uncertain prospects (acts de…ned
on a state space S) and serves as a component in the representation of that preference. We
begin by noting two critiques of the Savage model as a subjective theory of probability. Each
claims that the Savage model delivers ‘too much’to be completely satisfactory.
The …rst sense in which Savage delivers too much is that his axioms deliver not only
the fact that preference is based on probabilities, but also the expected utility functional
form. Because the use of probabilities seems more basic than any particular functional form,
this aspect of the Savage theory is unattractive as a theory of probability. This critique
is due to Machina and Schmeidler (1992) where it has been addressed by these authors
through axioms that deliver probabilistically sophisticated preferences. Roughly speaking,
probabilistic sophistication entails a two-stage procedure for evaluating any act. First, the
decision-maker uses a probability measure on the state space in order to translate the act into
an induced distribution over outcomes (a lottery); and second, she uses a (not necessarily
expected) utility function de…ned on lotteries to evaluate the induced lottery and produce a
utility level for the act. Thus preference is based on probabilities, but in a way that does
not impose super‡uous functional form restrictions.2
The second critique of the Savage theory that applies also to the Machina and Schmeidler
extension is more pertinent to this paper. Both theories deliver too much in that they derive
probabilities for all measurable events, that is, for all events in some prespeci…ed -algebra
(that could be the power set). Consequently, there is an important sense in which their
theories fail to be subjective. They are subjective in the sense that the probability measure on
that is delivered is derived from the decision-maker’s preference ordering over -measurable
acts. However, the domain of the measure is exogenous to the model rather than being
derived from preference.
Exogeneity of the -algebra is not a limitation of a ‘subjective theory’ if it is believed
that decision-makers assign probabilities to all events that are relevant to the context being
modeled. In that case, the modeling context may dictate the appropriate speci…cation for
2
That is not quite true as explained in Section 4.1.
2
, independently of preference. But choice behavior such as that exhibited in the Ellsberg
Paradox and related evidence have demonstrated that many decision-makers do not assign
probabilities to all events. In situations where some events are ‘ambiguous’, decision-makers
may not assign probabilities to those events, though the likelihoods of ‘unambiguous’events
are represented in the standard probabilistic way. For example, in the case of the Ellsberg
urn with balls of 3 possible colors, R, B and G, where the only objective information is that
R + B + G = 90 and R = 30, events in the class
A = f;; fRg,fB; Gg; fR; B; Ggg
(1.1)
are intuitively unambiguous. Most decision-makers would presumably assign them the obvious probabilities in deciding on how to rank bets based on the color of a ball to be drawn at
random. However, the use of probabilities for other events is inconsistent with the common
‘ambiguity averse’preference ranking of such bets, namely a preference to bet on R (drawing a red ball) rather than on B and also a preference for betting on fB; Gg rather than on
fR; Gg.
On the other hand, aversion to ambiguity is not universal. Some decision-makers are
indi¤erent to ambiguity and behave in the non-paradoxical and fully probabilistic fashion.
The lesson we take from this is that decision-makers may di¤er not only in the probabilities
assigned to given events (an aspect not well illustrated by this example), but also in the
identity of the events to which they assign probabilities. Thus to be fully subjective a theory
of probability should derive both the domain and the values of the probability measure from
preference.
The starting point for such a theory is the de…nition of ‘unambiguous’. Using our de…nition, we identify the class A of (subjectively) unambiguous events. Then we show (Theorem
5.2) that the decision-maker is probabilistically sophisticated on the domain of unambiguous
(A-measurable) acts, given suitable axioms on preference. This representation result constitutes a contribution towards a fully subjective theory of probability, because both the domain
A of the decision-maker’s probability measure and the values assigned by the measure to
events in A are derived from preference.
1.3. The Knightian Distinction
The Knightian distinction provides another perspective on the value of Theorem 5.2. Knight
used risk to refer to situations where (possibly subjective) probabilities apply and ambiguity
to refer to all other situations. Our de…nition of ambiguity also leads to the dichotomous
characterization of all ‘situations’, events or acts, as either unambiguous or ambiguous.
Then Theorem 5.2 delivers probabilities on the class of unambiguous events. In this way, it
provides precise expression to and a choice-theoretic foundation for the Knightian distinction
between risk and ambiguity. The resulting sharp distinction between (probabilistic) risk and
ambiguity also permits a uni…ed treatment of attitudes towards risk and attitudes towards
ambiguity, as described in Section 6.3
3
Schmeidler (1989) proposes a de…nition of ambiguity aversion for the case where preference is de…ned
over two-stage Anscombe-Aumann acts rather than merely over Savage-style acts as here. See Epstein (1999)
3
1.4. Desiderata
The preceding applications suggest some (overlapping) desiderata that guide our choice of
a de…nition of ambiguity. We seek a de…nition of ambiguity that, in addition to capturing
intuition, is:
D1 Behavioral or expressed in terms of preference: Following the choice-theoretic tradition
of Savage, we insist that ambiguity be de…ned in terms of preference. This ensures
veri…ability, at least in principle given suitable data on behavior, and hence potential
empirical relevance. The choice of a de…nition thus amounts to answering the question
“what behavior would indicate that the decision-maker perceives a speci…c event or
act as ambiguous?”
D2 Model-free: Ambiguity and attitudes towards ambiguity seem more basic than the use of
any particular functional form for utility. Thus to be satisfactory, a de…nition should
not be tied to particular models such as those in Schmeidler (1989) or Gilboa and
Schmeidler (1989).
D3 Explicit and constructive: Given an event, it should be possible to check whether or
not it is ambiguous. This would aid veri…ability. As well, an explicit de…nition would
ensure that the de…nition produces a unique class A of unambiguous events.4
D4 Consistent with probabilistic sophistication on unambiguous acts: We have mentioned
two reasons for seeking such a representation (as delivered by our Theorem 5.2). First,
it delivers a fully subjective theory of probability. Second, it delivers the Knightian
distinction between (probabilistic) risk and ambiguity, which in turn permits modeling, within a uni…ed framework, both attitudes towards risk and attitudes towards
ambiguity.
We provide a de…nition that performs fairly well in terms of these criteria. However,
two signi…cant limitations should be acknowledged at the outset. First, as explained in
Section 4.1, our de…nition makes sense only for preferences satisfying Savage’s axiom P 3.
Moreover, probabilistic sophistication on unambiguous acts is delivered only after assuming
some additional axioms. In our defense, assuming these axioms falls far short of assuming a
parametric model of preference such as Choquet expected utility or multiple-priors.
A second limitation that warrants emphasis concerns D1 and the subjective nature of
our theory of (un)ambiguity and probability. We de…ne ambiguity in terms of the preference
ranking of acts over an exogenously given state space (and outcome set). Thus the state
space and its associated universal -algebra of events are presumed ‘objective’. The way
in which our theory is more subjective than other theories is that we endogenize the subclass
of unambiguous events in . The importance of the assumed objectivity of S is that state
spaces and Savage-style acts are constructs used to model the choice of (physical) actions.
for a discussion of the limitations of Schmeidler’s de…nition and of applications that have been made of it.
4
See Section 8.2 for more on ‘uniqueness’.
4
Thus the empirically relevant question is “which actions are unambiguous?” We can answer
such a question only given a translation of actions into acts over some state space.5
1.5. Outline
The paper proceeds as follows. We conclude the introduction by describing a by-product
of the search for a fully subjective theory of probability that also casts light on technical
aspects of the paper. Next we de…ne the key notion of a -system and introduce some
notation. Section 3 introduces our de…nition of ambiguity and Section 4 describes axioms
that restrict the class of preferences that we admit. These axioms deliver our main result,
Theorem 5.2, in Section 5. The Theorem is applied in Section 6 to de…ne attitudes towards
risk and ambiguity and it is illustrated in Section 7 in the context of the Choquet expected
utility model. Our de…nition of ambiguity is contrasted with alternatives in Section 8 and
Section 9 provides further perspective on the de…nition by examining it in the context of
Ellsberg-style situations. Proofs are relegated to appendices.
1.6. A By-Product
Typically, probability theory posits that any probability measure is de…ned on an algebra or
-algebra, constructs that seem natural from a mathematical point of view. In a fully subjective theory, the domain A of the subjective probability measure, including its mathematical
properties, are derived. This permits the appropriateness of the standard assumptions to be
evaluated from a decision-theoretic point of view. This argument is due to Zhang (1997),
whose major …nding in this regard we proceed to outline.
The major point is that A is typically not a -algebra or even an algebra. Moreover, at
an intuitive level, while the class of unambiguous events is naturally taken to be closed with
respect to complements and disjoint unions, it may not be closed with respect to intersections.
This point may be illustrated by borrowing Zhang’s example of an Ellsberg-type urn with
4 possible colors - R; B; G and W . Suppose that the only objective information is that the
total number of balls is 100 and that R + B = G + B = 50. Then it is intuitive that the
class of unambiguous events is
A = f S; ;; fB; Rg; fB; Gg; fG; W g; fR; W gg.
(1.2)
Observe that A fails to be an algebra, because while fB; Rg and fB; Gg are unambiguous,
their intersection fBg is not. As pointed out by Zhang, the appropriate mathematical
structure for A is a -system (de…ned below), also sometimes called a Dynkin system in the
measure theory literature.
For this paper, the fact that we cannot take A to be an algebra complicates the derivation
of a probability measure on A and, in particular, prevents us from simply invoking existing
results from Savage (1954), Fishburn (1970) and Machina and Schmeidler (1992). The
5
It is an open question whether two representations of the same choice environment, using di¤erent state
spaces and preferences satisfying our axioms, can lead to di¤erent conclusions about ambiguity.
5
arguments in these studies exploit the fact that the relevant class of events is closed with
respect to intersections. We rely instead on a recent representation result in Zhang (1999)
for qualitative probabilities on -systems.
2. PRELIMINARIES
Let (S; ) be a measurable space where S is the set of states and is a -algebra. All events
in this paper are assumed to lie in ; we repeat this explicitly below only on occasion.
Say that a nonempty class of subsets A
of S is a –system if
.1 S 2 A;
.2 A 2 A =) Ac 2 A; and
.3 An 2 A; n = 1; 2; ::: and Ai \ Aj = ;; 8i 6= j =) [1
n=1 An 2 A.
This de…nition and terminology appear in Billingsley (1986). A –system A is closed with
respect to complements and countable disjoint unions. The intuition for these properties is
clear if we think of A as a class of events to which the decision-maker attaches probabilities.
If she can assign a probability to event A, then the complementary probability is naturally
assigned to Ac : Similarly, if she can assign probabilities to each of the disjoint events A and B,
then the sum of these probabilities is naturally assigned to A [ B. On the other hand, there
is no such intuition supporting closure with respect to intersections, or equivalently, with
respect to arbitrary unions. Lack of closure with respect to intersections di¤erentiates –
systems from algebras or –algebras. As illustrated by the above example of an Ellsberg-type
urn with 4 colors, –systems are more appropriate for modeling families of ‘unambiguous’
events.
We have the following equivalent de…nition:6
Lemma 2.1. A nonempty class of subsets A
of S is a –system if and only if
.10 ;; S 2 A;
.20 A; B 2 A and A
.30 An 2 A and An
B =) BnA 2 A; and
An+1 ; n = 1; 2; :::; =) [1
n=1 An 2 A.
Although A is not an algebra, a probability measure can still be de…ned on A. Say that
p : A 7 ! [0; 1] is a (…nitely additive) probability measure on A if
P.1 p(;) = 0; p(S) = 1; and
6
See Billingsley (1986, p. 43). Collections of sets satisfying the following conditions are frequently called
alternatively Dynkin systems or d-systems (see, for example, Williams (1991, p. 193)).
6
P.2 p(A [ B) = p(A) + p(B); 8A; B 2 A; A \ B = ;:
Countable additivity of p is de…ned in the usual way and will be stated explicitly where
needed. Given a probability measure p on A, call p convex-ranged if for all A 2 A and
0 < r < 1, there exists B A, B 2 A, such that p(B) = r p(A).7
As in Savage, we assume a set of outcomes X . Prospects are modeled via (simple) acts, measurable maps from S to X having …nite range. The set of acts is F = f:::; f; f 0 ; g; h; :::g.
Given a -system A, de…ne F ua by
F ua = ff 2 F : f is A-measurableg,
(2.1)
where f is A-measurable if fs 2 S : f (s) 2 Xg 2 A for any subset X of X . Thinking of A
as the set of unambiguous events, F ua is naturally termed the set of unambiguous acts.
3. UNAMBIGUOUS EVENTS
3.1. Ellsberg-Based Intuition
As suggested in the context of the …rst desideratum, a de…nition of ambiguity must answer
the question “which behavior would reveal that the decision-maker views a given event as
‘ambiguous’?” The Ellsberg urn with three colors illustrates our approach. The typical
choices described earlier take the form
0
1
0
1
100 if s = R
0
if s = R
@ 0
@ 100 if s = B A and
if s = B A
(3.1)
0
if s = G
0
if s = G
0
1
0
1
100 if s = R
0
if s = R
@ 0
@ 100 if s = B A .
if s = B A
100 if s = G
100 if s = G
Thus the preference to bet on red rather than blue is reversed by the change in the outcome
associated with a green ball. The intuition for the reversal is the complementarity between
G and B - there is imprecision regarding the likelihood of B whereas fB; Gg has precise
probability 2/3. As a result, the change to 100 if G (drawing a green ball) provides an
entirely di¤erent perspective on the relative valuation of 100 if R as opposed to 100 if B. In
alternative notation that renders the complementarity more transparent,
R
where
`
`
B but R [ G
`
B [ G,
is interpreted as ‘would rather bet on’.
7
When A is a -algebra and p is countably additive, this is equivalent to non-atomicity (Rao and Rao
(1983, pp. 142-3)).
7
We take such nonseparability as indicative of ambiguity. More precisely, we view such a
reversal in rankings as the behavioral manifestation of the intuitively ambiguous nature of
the event G and we use it as the basis for our de…nition of ambiguity in a general setting.
In fact, the above intuition is more general than may be apparent from (3.1). To see
this, expand the urn to contain 150 balls in total. There are still 30 red and 60 that are
either blue or green but in addition, there are 60 balls that are either white or yellow. The
decision-maker may be given some (possibly perfect) information about the relative numbers
of white and yellow balls. Then the following rankings are intuitive for reasons similar to
those underlying (3.1):
0
1
0
1
100 if s = R
0
if s = R
B 0
C
B 100 if s = B
C
if s = B
B
C
B
C
(3.2)
@ h(s) if s 2 fW; Y g A
@ h(s) if s 2 fW; Y g A and
0
if s = G
0
if s = G
0
100
B 0
B
@ h(s)
100
if
if
if
if
0
1
s=R
C
s=B
C
s 2 fW; Y g A
s=G
0
B 100
B
@ h(s)
100
if
if
if
if
1
s=R
C
s=B
C,
s 2 fW; Y g A
s=G
where the prizes h(W ) and h(Y ) for drawing white and yellow balls are arbitrary (but
common to all 4 acts). Such a reversal of rankings, which once again admits interpretation as
a form of nonseparability, is a behavioral manifestation of the intuitive ambiguity of G in the
context of the present modi…ed Ellsbergian setting with enlarged state space fR; B; G; W; Y g.
One …nal point of clari…cation concerns the fact that the story we have just told suggests
that the ambiguity of G leads to preference reversals as above for all subacts h. This is the
case only because of our (implicit) assumption that the added and unspeci…ed information
about W and Y does not also provide further information about B and G. In such a case,
complementarities between the two pairs of colors may be such that the preference reversal
would not occur for all acts h, but it would still occur for some h.8 This ‘explains’completely
the de…nition to follow.
3.2. De…nition
The primitives (S; ) and X are de…ned as above. The decision-maker has a preference order
on the set of acts F. Unambiguous events are now de…ned from the perspective of .
De…nition: An event T is unambiguous if: (a) For all disjoint subevents A; B of T c , acts
8
If the added information is that W = B, then the reversal of rankings is intuitive as above if h( ) = 0.
However, it is plausible that for h(W ) = 100 and h(Y ) = 0, the …rst ranking in (3.2) is satis…ed and this
ranking is not a¤ected by the change in outcome from 0 to 100 on G.
8
h and outcomes
0
x
B x
B
@ h(s)
z
0
x
B x
B
@ h(s)
z0
x ; x; z; z 0 2 X ,
if
if
if
if
if
if
if
if
1
s2A
C
s2B
C
s 2 T c n(A [ B) A
s2T
0
x
B x
B
@ h(s)
z
if
if
if
if
s2A
C
s2B
C
s 2 T c n(A [ B) A
s2T
x
B x
B
@ h(s)
z0
if
if
if
if
1
0
1
s2A
C
s2B
C
s 2 T c n(A [ B) A
s2T
1
=)
(3.3)
s2A
C
s2B
C;
s 2 T c n(A [ B) A
s2T
and (b) The condition obtained if T is everywhere replaced by T c in (a) is also satis…ed.
Otherwise, T is ambiguous.
The set of unambiguous events is denoted A. It is nonempty because ; and S are unambiguous. Observe that the de…ning invariance condition is required to be satis…ed even if A or B
is empty. The requirement that both T and T c satisfy the indicated invariance builds into
the de…nition the intuitive feature that an event is unambiguous if and only if its complement
is unambiguous.
Turn to further interpretation, assuming that x
x.9 The …rst two acts being compared
yield identical outcomes if the true state lies in (A [ B)c . Thus the comparison is between
‘bets conditional on (A [ B)’with stakes x and x and the outcomes shown for (A [ B)c . The
indicated ranking reveals that the decision-maker views A as conditionally more likely than
B. Suppose now that the outcome on event T is changed from z to z 0 . If T is ‘unambiguous’,
then this conditional likelihood ranking should not be a¤ected because ‘unambiguous’means
or at least entails such separability or invariance.
A moment’s re‡ection on this intuition may help to alleviate concerns regarding features
of the de…nition that may seem arbitrary. After all, why is it the case that acts are restricted
to be constant within each of the events A, B and T , even though outcomes are allowed to
vary within T c n(A[B)? In fact, these restrictions are vital for the preceding intuition. First,
the interpretation in terms of the invariance of the relative conditional likelihoods of A and
B is justi…ed only because of the constancy of outcomes on each of A and B. More general
comparisons of the form
0
1
0
1
f (s) if s 2 A
g(s) if s 2 A
B g(s) if s 2 B
C
B f (s) if s 2 B
C
B
C
B
C
vs
@ h(s) if s 2 T c n(A [ B) A
@ h(s) if s 2 T c n(A [ B) A
z
if s 2 T
z
if s 2 T
re‡ect not only assessments of conditional likelihoods but also other aspects of preference
such as attitudes towards uncertainty (risk plus ambiguity).10 Thus invariance of rankings
9
Any x 2 X denotes both the outcome and the constant act producing the outcome x in every state.
Thus ‘x
x’has the obvious meaning.
10
For elaboration see the parallel discussion in Machina and Schmeidler (1992, Section 4.2)).
9
of this sort has nothing apparent to do with the ambiguity of T .
Similarly, constancy of acts on T is vital for the above intuition. It is T in its entirety
that is unambiguous and this does not imply anything about subsets. This leads naturally
to the restriction to acts that are constant on T .
On the other hand, there is no good reason to restrict outcomes within T c n(A [ B). The
previous subsection illustrated the intuitive plausibility of requiring invariance even when
outcomes vary within T c n(A [ B). More to the point, we are not aware of any intuition for
a de…nition with h restricted to be constant that does not also suggest that the indicated
invariance should hold for all h.
A similar point is relevant to another question that may have occurred to some readers.
Though in the motivating Ellsberg-style examples, the outcomes z and z 0 equal either x or
x, the intuition just described applies equally to general z and z 0 .11 Moreover, this generality
is important below in that (our proof of) Theorem 5.2 exploits it.
Conclude with two brief observations regarding the intuitive performance of the de…nition.
First, all events are unambiguous if is probabilistically sophisticated, a fortiori if is a
subjective expected utility order.12 Second, in the motivating example of an Ellsberg urn
with 3 colors, the events fGg and fBg are subjectively ambiguous for any preference order
that predicts the typical choices (3.1). (The designation of fRg and other events depends
on more information about preference than is contained in (3.1).) See Section 9 for other
Ellsberg-type settings.
4. AXIOMS
Though the de…nition seems to capture some aspects of ambiguity, our proposal is that it
be adopted only subject to some arguably weak axioms for preference that are speci…ed in
this section. These axioms serve both to strengthen intuitive appeal (this is the primary role
of the …rst axiom, Savage’s P 3) and to satisfy desideratum D4 by delivering probabilistic
sophistication on the domain of unambiguous acts (this is the role of the remaining axioms).
4.1. Savage’s P3, Separability and Ambiguity
With regard to intuitive appeal, a natural concern regarding our de…nition is that the nonseparability re‡ected by a violation of the invariance in (3.3) may fail for reasons that have
nothing to do with ambiguity. To see this, consider Savage’s axiom P 3:13
Axiom 1. (Savage’s P 3): For all non-null events A in
if and only if (x if A; f ( ) if Ac )
(x if A; f ( ) if Ac ).
11
and for all x , x and f , x
x
More elaborate examples are readily constructed to illustrate this.
That A =
given probabilistic sophistication, follows immediately from Machina and Schmeidler’s
central axiom P 4 .
13
The event A is null (with respect to ) if f 0
f whenever f 0 ( ) = f ( ) on SnA.
12
10
The axiom imposes a form of monotonicity and also state independence (the preference for
x over x is independent of the common event where they are realized). The necessity of
this axiom in our approach implies, in particular, that we have nothing to say about the
meaning of ambiguity when preferences are state dependent.
One connection between P 3 and our de…nition of ambiguity is apparent on examination
of the intuition provided following the de…nition. That intuition, regarding the invariance
of relative conditional likelihoods, implicitly assumes that if x
x, then x would also be
weakly preferred conditionally in the sense that, for any act f ,
(x if A; f ( ) if Ac )
(x if A; f ( ) if Ac ) ,
which is one direction in P 3.
Another connection stems from the fact that P 3 implies: For all non-null A and for all
x , x, f and g,
(x if A; f ( ) if Ac )
(x if A; f ( ) if Ac ) ()
(4.1)
(x if A; g( ) if Ac )
(x if A; g( ) if Ac ) .
In other words, rankings are invariant to changes in common outcomes for the comparisons
shown, which admits interpretation as a form of separability across outcomes. Therefore,
if in (3.3) the switch from outcome z to z 0 on T causes a reversal of rankings, ‘then’ this
reversal is due to complementarities between events (as in the intuition suggested for our
de…nition) rather than between outcomes (the noted switch on T causing a change in the
relative evaluation of x versus x).
For the above reasons, our de…nition is appropriate only for preference orders satisfying
P 3, which we assume in the sequel. We have evidently not succeeded in capturing ambiguity for all preference orders and desideratum D2 is not ful…lled. However, P 3 is a common
and arguably mild assumption and is much weaker than imposing a parametric model of
preference. In addition, it is useful for perspective to note that there is a parallel limitation associated with the notion of probabilistic sophistication. As de…ned by Machina and
Schmeidler (and as adopted below), probabilistic sophistication captures not only the reliance of preference on probabilities but also monotonicity as embodied in P 3, which axiom
is clearly not germane to the use of probabilities.14
The role of P 3 is partly interpretational, but it also plays a role in the formal analysis
(Theorem 5.2). We turn next to other axioms that play a formal role.
4.2. Further Axioms
The further preference axioms speci…ed here will deliver not only the -system properties
for A and a probability measure on these unambiguous events, but also the probabilistic
sophistication of preference restricted to unambiguous acts. Reasons for seeking such a
representation result were o¤ered in the introduction.
14
The generalization by Grant (1995) also assumes some monotonicity, though a weaker form.
11
The set F ua of unambiguous acts is de…ned by (2.1). Also useful, for any given A 2 A,
is the set of acts
FAua = ff 2 F : f
1
(X) \ A 2 A for all X
X g.
Some of the axioms for are slight variations of Savage’s axioms, with names adapted
from Machina and Schmeidler. Though the axioms are expressed in terms of A, they constitute assumptions about because A is derived from . A …nal remark is that the remaining
axioms relate primarily to restricted to F ua .
Axiom 2. (Nondegeneracy): There exist outcomes x and x such that x
x.
Axiom 3. (Weak Comparative Probability): For all events A; B 2 A and outcomes
x
x and y
y
x
x
if s 2 A
if s 2 Ac
x
x
if s 2 B
if s 2 B c
y
y
if s 2 A
if s 2 Ac
y
y
if s 2 B
if s 2 B c
()
:
This is Savage’s axiom P 4 restricted to unambiguous events. It requires that for unambiguous
events A and B, the preference to bet on A rather than on B is independent of the stakes.
The axiom delivers the complete and transitive likelihood relation ` on A, where A ` B
if 9x
x such that
x
x
if s 2 A
if s 2 Ac
x
x
if s 2 B
if s 2 B c
.
The next axiom imposes suitable richness of the set of unambiguous events. It is clear
from Savage’s analysis that some richness is required to derive a probability measure on A.
Savage’s axiom P 6 (suitably translated) is not adequate here because A is not a -algebra.
However, the spirit of Savage’s P 6 is retained in the next axiom.
Axiom 4. (Small Unambiguous Event Continuity): Let f; g 2 F ua , f
g, with
f = (x1 ; A1 ; x2 ; A2 ; :::; xn ; An ), g = (y1 ; B1 ; y2 ; B2 ; :::; ym ; Bm ), where each Ai and Bi lies
M
in A. Then for any x in X , there exist two partitions fCi gN
i=1 and fDj gj=1 of S in A that
re…ne fAi gni=1 and fBj gm
j=1 respectively, and satisfy:
f
x
if s 2 Dk
g(s) if s 2 Dkc
; for all k 2 f1; :::; N g;
(4.2)
g; for all j 2 f1; :::; M g:
(4.3)
and
x
if s 2 Cj
f (s) if s 2 Cjc
12
Very roughly, the axiom requires that unambiguous events can be decomposed into suitably
‘small’unambiguous events. When A is closed with respect to intersections, as in the Savage
or Machina-Schmeidler models where it is taken to be the power set, then the axiom is implied
by Savage’s P 6, given Axioms 1-3.15
The preceding axioms are largely familiar, at least when imposed on all of F, rather than
just on F ua as here. The remaining two axioms are ‘new’and are needed to accommodate
the fact that A may not be a -algebra.
ua
Say that a sequence ffn g1
converges in preference to f1 2 F ua if: For any two
n=1 in F
acts f ; f in F ua satisfying f
f1 f , there exists an integer N such that
f
fn
f , whenever n
N:
Axiom 5. (Monotone Continuity): For any A 2 A, outcomes x
x, act h 2 FAuac and
decreasing sequence fAn g1
A, de…ne
n=1 in A with A1
0
1
0
1
x
if s 2 An
x
if s 2 \1
n=1 An
A.
if s 2 AnAn A and f1 = @ x
if s 2 An(\1
fn = @ x
n=1 An )
c
c
h(s) if s 2 A
h(s) if s 2 A
ua
If fn 2 F ua for all n = 1; 2; :::, then ffn g1
n=1 converges in preference to f1 and f1 2 F .
The name Monotone Continuity describes one aspect of the axiom, that requiring the indicated convergence in preference.16 The second component of the axiom is the requirement
that the limit f1 lie in F ua whenever each fn is unambiguous. This will serve in particular to
ensure that A satis…es the ‘countable’closure condition :3 or :30 required by the de…nition
of a -system.
It might be felt that given the correct de…nition of ‘unambiguous’, the derivation of a
probability measure on A should be possible with little more than some richness requirements. The axioms stated thus far can arguably be interpreted as constituting such minimal
requirements. However, they do not su¢ ce and we need one …nal axiom. This may re‡ect the
fact that only some aspects of ‘unambiguous’are captured in our de…nition. In any event,
the …nal axiom is intuitive and arguably weak. Its statement requires some preliminaries.
A …nite partition with component events from A is denoted fAi g. Henceforth all partitions have unambiguous components, even where not stated explicitly. Given such a partition, use the obvious abbreviated notation (xi ; Ai ). For any permutation of f1; :::; ng,
x (i) ; Ai denotes the act obtained by permuting outcomes between the events. Say that the
15
Savage’s P 6 applied to F ua would require that, given x, if f = (xi ; Ai )ni=1
g = (yi ; Bi )ni=1 , where
N
every Ai and Bi lies in A, then there exists a partition fGi gi=1 of S in A such that f
(x; Gk ; g; Gck ) for
all k. Given such a partition, and given that A is closed with respect to intersections, then the collection of
events Dij = Gi \ Bj satisfes the requirements in Axiom 4.
16
A related axiom with the same name is used by Arrow (1970) to deliver the countable additivity of the
subjective probability measure. Here as well, countable additivity will follow from Monotone Continuity, but
as an unintentional by-product.
13
…nite partition fAi g is a uniform partition if Ai ` Aj for all i and j and call fAi g strongly
uniform if in addition it satis…es: For all outcomes fxi g and for all permutations ,
x
(xi ; Ai ) .
(i) ; Ai
(4.4)
In particular, if fAi gni=1 is a strongly uniform partition, then for all index sets I and J,
subsets of f1; 2; :::; ng,
[i2I Ai ` [i2J Ai if j I j = j J j .
Axiom 6. (Strong-Partition Neutrality): For any two strongly uniform partitions fAi gn1
and fBi gn1 , if Ai ` Bi for all i, then for all fxi g,
0
1 0
1
x1 if s 2 A1
x1 if s 2 B1
B x2 if s 2 A2 C B x2 if s 2 B2 C
B
C B
C.
(4.5)
@ ... ...
A @ ... ...
A
xn if s 2 An
xn if s 2 Bn
The hypothesis that the Ai ’s and Bi ’s satisfy (4.4) expresses another sense in which these
events are unambiguous. This makes the conclusion (4.5) natural and much weaker than if
the indi¤erence in (4.5) were required for all uniform partitions. The latter axiom would go a
long way towards explicitly imposing probabilistic sophistication, an unattractive feature in
the present exercise where the intention is that probabilistic sophistication on unambiguous
acts should result primarily from the de…nition of ‘unambiguous’. Axiom 6 is less vulnerable
to such a criticism.
To support the claim that Strong-Partition Neutrality is a relatively weak axiom, observe
that it is satis…ed by all Choquet-expected-utility functions as de…ned in Section 7, proving
that it falls far short of imposing probabilistic sophistication.17
We turn next to the implications of these axioms.
5. PROBABILISTIC SOPHISTICATION ON UNAMBIGUOUS
ACTS
De…ne “probabilistic sophistication on unambiguous acts F ua ”by extending the de…nition of
Machina and Schmeidler. Their de…nition is obtained if one replaces F ua and A by F and
respectively in the formulation to follow. It is occasionally distinguished here terminologically
by referring to “global probabilistic sophistication.”
Some preliminary notions are required. Denote by D(X ) the set of probability distributions on X having …nite support. A probability distribution P = (x1 ; p1 ; :::; xm ; pm ) is said
to …rst-order stochastically dominate Q = (y1 ; q1 ; :::; yn ; qn ) with respect to the order over
the outcome set X if
X
X
pi
qj
for all x 2 X .
fi:xi xg
fj:yj xg
17
If fAi g is a strongly uniform partition, then the capacity
by the partition. Thus the indi¤erence (4.5) is implied.
14
must be additive on the algebra generated
Use the term strict dominance if the above holds with strict inequality for some x 2 X .
Given a real-valued function W de…ned on a mixture subspace dom(W ) of D(X ), say
that W is mixture continuous if for any distributions P , Q and R in dom(W ), the sets
f 2 [0; 1] : W ( P + (1
f 2 [0; 1] : W ( P + (1
)Q)
)Q)
W (R)g and
W (R)g
are closed. Say that W is monotonic (with respect to stochastic dominance) if W (P )(>)
W (Q) whenever P (strictly) stochastically dominates Q, P and Q in dom(W ).
Given a probability measure p on A, denote by Pf;p 2 D(X ) the distribution over outcomes induced by the act f . De…ne
Dpua (X ) = fPf;p : f 2 F ua g.
When p is convex-ranged, then Dpua (X ) is a mixture space.
We can …nally state the desired de…nition. Say that is probabilistically sophisticated on
ua
F if there exists a convex-ranged probability measure p on A and a real-valued, mixture
continuous and monotonic function W on Dpua (X ) such that has utility function U of the
form
U (f ) = W (Pf;p ) .
(5.1)
Roughly speaking, the probability measure p is used to translate acts in F ua into (purely
risky) lotteries and these are evaluated by means of the risk preference functional W . No
stand is taken on the functional form of W , apart from monotonicity and mixture continuity,
thus capturing primarily the decision-maker’s reliance on probabilities for the evaluation of
unambiguous acts. Subjective expected utility is merely one example, albeit an important
one, in which W is an expected utility function on lotteries Dpua (X ) and thus U has the
familiar form
Z
U (f ) =
u(f ) dp.
(5.2)
S
We remind the reader that because A and F ua are derived from the given primitive preference relation on F, probabilistic sophistication so-de…ned is a property of exclusively
and does not rely on an exogenous speci…cation of ‘unambiguous acts.’
Probabilistic sophistication with measure p implies that likelihood (or the ranking of
unambiguous bets) is represented by p; that is,
A
`
B
()
pA
pB, for all A; B 2 A.
But (5.1) is much stronger, requiring that the ranking of all (not necessarily binary) unambiguous acts be based on p.
Turn to the implications of our axioms. A preliminary result (proven in Appendix A) is
that they imply that A is a -system.
Lemma 5.1. Under Axioms 2, 4 and 5, A is a -system.
15
The following is our main result:
Theorem 5.2. Let
be a preference order on F satisfying P 3 and A the corresponding
set of unambiguous events. Then the following two statements are equivalent:
(a)
satis…es axioms 2-6.
(b) A is a -system and there exists a (unique) convex-ranged and countably additive
probability measure p on A such that is probabilistically sophisticated on F ua with
underlying measure p.
The bulk of the proof is found in Appendices B and C. The arguments used by Savage,
Fishburn and Machina-Schmeidler must be modi…ed because only in the present setting is
the relevant class of events A not closed with respect to intersections. A key step is to show
(Appendix B) that our axioms for preference deliver the conditions for the implied likelihood
relation that are used by Zhang (1999) in order to obtain a representing probability measure.
The proof of probabilistic sophistication is completed in Appendix C.
The …rst step in discussion of the theorem is a comparison with the main result in
Machina and Schmeidler (1992). If their axioms are imposed, then all events are subjectively
unambiguous.18 Therefore, one can view their result as a special case where A =
and
19
global probabilistic sophistication prevails. Here, in contrast, A is allowed to be smaller
than , in a way that depends on preference . This relaxation renders Theorem 5.2 a
contribution towards a fully subjective theory of probability as discussed in the introduction.
Also claimed there (and summarized in desideratum D4) is that our representation result
provides foundations for the Knightian distinction between probabilistic risk, represented by
prospects in F ua , and ambiguity, represented by all other acts. In Section 6, we exploit this
aspect of the theorem to describe a uni…ed treatment of attitudes towards each of risk and
ambiguity.
A noteworthy feature of our theorem is that it is silent on the nature of preference on
the domain of ambiguous acts. While at …rst glance this may seem like a weakness, we feel
to the contrary that it is a strength.20 As argued in the introduction, a theory of probability
should not deliver structure, such as the expected utility functional form restriction, that
is not germane to the use of probabilities. In this sense, restrictions on the ranking of
ambiguous acts constitute excess baggage and it is a virtue of our representation theorem to
have avoided them (apart from P 3).
It is decidedly not the case that the noted silence renders the theorem irrelevant to the
analysis of choice when there are ambiguous acts. The contrary is true. As just mentioned
and as described further in Section 6, the theorem provides the basis for a uni…ed and
coherent treatment of risk aversion and ambiguity aversion. It is precisely because our axioms
18
See the end of Section 3.2.
A quali…cation is that our theorem delivers a countably additive probability measure, while theirs deals
with the more general class of …nitely additive subjective priors.
20
We are indebted to Michelle Cohen and Mark Machina for helping to clarify our thinking on this point.
19
16
do not restrict the structure of preference over ambiguous acts that the scope of the theorem
is not unduly limited. The theorem and the use we make of it apply to the Choquet and
multiple-priors models and to any other model of choice among ambiguous acts, subject only
to the satisfaction of our axioms regarding (for the most part) the ranking of unambiguous
acts.
Finally, consider another possible route to deriving from preference both domains and
probability measures on those domains.21 Simply identify domains A where the MachinaSchmeidler axioms, (suitably modi…ed to allow for the fact that A may be only a -system
and not an algebra), are satis…ed and apply their result to derive corresponding probabilities.
Evidently, such a process delivers probabilistic sophistication on an endogenous subdomain
F of acts. One might view acts in F as risky and use them to support the Knightian distinction. A problem is that in general there will be several such domains A and F without
any guidance for selecting the right one as the foundation for the Knightian distinction. Our
approach makes such a selection and one that has the merit of being based on an intuitive
and behavioral characterization of ambiguity.
Another point of comparison is that we are able to deliver a probability measure on A
with relatively weak axioms. In particular, we do not assume the Sure-Thing-Principle on
F ua nor that the counterpart axiom used by Machina and Schmeidler is satis…ed on F ua .
We are able to avoid assuming these because, unlike these earlier authors, we are interested
in deriving probabilities only on unambiguous events and because ‘unambiguous’ has been
de…ned in an appropriate way.
6. ATTITUDES TOWARDS RISK AND AMBIGUITY
Once one accepts the distinction between risk and ambiguity as meaningful, it follows that
attitudes towards risk and towards ambiguity constitute conceptually distinct aspects of
preference. Therefore, one would like a modeling approach that a¤ords a clear distinction
between them. The key to the latter is the sharp distinction between risk and ambiguity
that is provided by Theorem 5.2. In this section, we show how the theorem leads to natural
de…nitions for attitudes towards risk and ambiguity.22
Consider …rst risk attitudes.23 The prior question is what constitutes risk? From Theorem
5.2, unambiguous acts are probabilistic prospects and it would seem natural to de…ne risk
attitudes in terms of the ranking of these acts. Moreover, given that is probabilistically
sophisticated on F ua , with unique probability measure p on A, the familiar de…nition can
21
This approach is discussed further in Section 8.
The approach is adapted from Epstein (1999), where it is discussed at greater length. The di¤erence here
is that the reference class of unambiguous events is derived from preference, while it is speci…ed exogenously
by the modeler in the cited paper. The latter study discusses also the comparative notions ‘more risk averse
than’and ‘more ambiguity averse than’. Given the length of this paper, we do not pursue these here.
23
For what follows assume that X is a convex subset of Rn .
22
17
be adapted. Therefore, say that is risk averse if
Z
h(s) dp
h, for all h in F ua ,
(6.1)
S
where the vector-valued integrals are interpreted as constant acts. Risk loving is de…ned by
reversing the direction of preference; risk neutrality is the conjunction of risk aversion and
risk loving.
Implicit in this de…nition of risk aversion are two normalizations: (i) the identi…cation of
riskless prospects with degenerate lotteries or constant acts, and (ii) the identi…cation of riskneutrality with expected value preferences. A de…nition of ambiguity aversion (or a¢ nity)
can be formulated in a similar fashion once one adopts a parallel pair of normalizations
that specify (i) unambiguous prospects and (ii) ambiguity neutral preferences. Our analysis
suggests a natural choice for each - acts in F ua constitute the class of unambiguous prospects
and (globally) probabilistically sophisticated preferences de…ne the benchmark of ambiguity
neutrality.
This leads to the following de…nition: Say that the preference order on F is ambiguity
averse if there exists another order ps on F , that is probabilistically sophisticated there,
such that
h ps ( ps ) f =) h ( ) f ,
(6.2)
for all h in F ua and f in F. Here F ua denotes the set of subjectively (for ) unambiguous acts
as de…ned in this paper. The interpretation begins with the view that any probabilistically
sophisticated order ps is indi¤erent to ambiguity. Accordingly, if ps prefers h to f , then so
should the ambiguity averse , because h is unambiguous for while discounts f further
due to its being ambiguous.
Ambiguity loving is de…ned by reversing the directions of preference in (6.2); ambiguity
neutrality is the conjunction of ambiguity aversion and ambiguity loving.
Because of the subjective nature of ambiguity, one might expect a confounding between
the absence of ambiguity on the one hand, and the presence of ambiguity combined with
indi¤erence to it on the other. The source of such confounding in our approach is in the
identi…cation of probabilistically sophisticated preference as ambiguity neutral even though
probabilistically sophisticated decision-makers do not perceive any ambiguity. There is a
similar confounding of the absence of risk with indi¤erence to it. If A = f;; Sg, then only
constant acts are unambiguous and perceived as risky. Yet such a decision-maker is deemed
to be risk neutral.24
However, a more important separation is a¤orded by our model, namely that between
attitudes towards risk and attitudes towards ambiguity. Informally, such a separation is
possible because risk attitudes concern the nature of preference within F ua , while attitudes
towards ambiguity concern primarily the way in which acts in F ua are related to acts outside
F ua (see (6.2)). The separation that is achieved is the following: For each of risk and
ambiguity, the following three (exhaustive but overlapping) attitudes are possible - aversion,
24
Strictly speaking, this extreme is ruled out by our axioms.
18
a¢ nity and ‘none of the above’(that is, neither aversion nor a¢ nity holds globally). All 9
logical possibilities are admitted by our de…nitions; for example, it is possible to model a
decision-maker who loves risk but dislikes ambiguity, and so on. This is amply demonstrated
by the Choquet-expected-utility model to which we now turn (see Corollary 7.4).
7. CHOQUET EXPECTED UTILITY
Here we focus on preference that conforms to Schmeidler’s (1989) Choquet expected utility
(CEU) model. This is one of the major alternatives to subjective expected utility theory in
the context of ‘ambiguity’. Therefore, it is appropriate to illustrate our de…nitions and to
demonstrate their tractability in the context of this speci…c model. We have much less to
say for the multiple-priors model and therefore we con…ne ourselves to CEU.
For CEU, preference is represented by U ceu , where25
Z
ceu
U (f ) =
u(f ) d .
(7.1)
S
Here :
! [0; 1] is a capacity and u : X ! R1 , with u(X ) having nonempty interior.
Preference so de…ned satis…es P 3 if and only if satis…es: For all disjoint events E and B,
(E [ B) = (B) ()
(7.2)
(E) = 0.
De…ne core( ), the core of , as in co-operative game theory, that is, as the set of all
…nitely additive probability measures p on S satisfying
p(A)
Say that
(A), for all A
S.
is exact if
(A) =
min
m2core( )
m(A), for all A
A stronger property that is widely assumed is convexity, where
(A [ B) + (A \ B)
(7.3)
S.
is convex if
(A) + (B),
for all events A and B.26
A special case of CEU preferences that is particularly relevant here has
=
25
(p),
(7.4)
is a capacity if it maps
into [0; 1], (E 0 )
(E) whenever E 0
E, (;) = 0 and (S) = 1.
The indicated integral is a Choquet integral and equals ni=1 ui [ ([nj=i Ej )
([nj=i+1 Ej )] if Ei = fs :
u(f (s)) = ui g and u1 < ::: < un .
26
See Schmeidler (1972) for the relation between convexity and exactness. Note that the CEU function
(7.1) is a member of the multiple-priors class if and only if is convex.
19
for some probability measure p and some strictly increasing
: [0; 1] ! [0; 1]. The
resulting preference order is probabilistically sophisticated and thus all events in are unambiguous in this case, that is, A = . (Probabilistic sophistication follows by verifying
that the risk preference functional W required by the de…nition in Section 5 can be taken to
be
Z
W (P ) =
u d ( (FP )) ,
X
for any lottery P , where FP is the cumulative distribution function corresponding to the probability measure P on X . Such functionals W correspond to the rank-dependent-expectedutility model that has been studied in the theory of preference over lotteries.27 )
For general capacities, the characterization of A is of interest. On purely formal (or
mechanical) grounds, thinking of the capacity as analogous to a probability measure, albeit
non-additive, each of the following classes of events seems plausible as a conjecture for how
to characterize unambiguous events:
A0 = fT
S : (T + A) = T + A for all A
A1 = fT
T c g,
S : T + T c = 1g and
A2 = \m2core( ) fA
S : mA = Ag.
The …rst two classes may seem natural because they capture forms of additivity of the
capacity. The class A2 consists of those events on which all measures in the core of agree,
which also seems to re‡ect a lack of ambiguity.
In general, A0
A1 . When is exact, these three sets coincide (see Ghirardato and
Marinacci (1999)).
Lemma 7.1. If
is exact, then A0 = A1 = A2 .
However, none of the above classes coincides with the class A of subjectively unambiguous
events. This is illustrated starkly by taking to be a distortion of a probability measure as
in (7.4), with the distortion being strictly convex. Then28
A =
and A0 = A1 = A2 = f;; Sg.
(7.5)
The next lemma provides a complete characterization of A (see Appendix D for a proof).29
Lemma 7.2. T is unambiguous if and only if: For all events A and B contained in T c and
for all C
T c n(A [ B),
(A)
(B) () (A [ T )
27
(B [ T ); and
(7.6)
See Chew, Karni and Safra (1987).
Under these assumptions, is a convex capacity, hence exact.
29
The characterizing conditions on the capacity may appear ugly; they are more complex than those used
to de…ne A0 and A1 , for example. However, it is clearly the appeal of the underlying behavioral de…nition
of ambiguity that is the critical esthetic factor.
28
20
if ( (A)
(A)
(A [ C)
(B)) ( (A [ C)
(B) =
(A [ T )
(B [ C)) < 0, then
(7.7)
(B [ T ) and
(7.8)
(B [ C) = (A [ C [ T )
and the above conditions are satis…ed also by T c .
(B [ C [ T ) ;
(7.9)
If the ‘reversal’in (7.7) never occurs, then is (almost) a qualitative probability within T c .30
In that case, the ordinal condition (7.6) alone corresponds to ‘T unambiguous’. However,
when fails to be a qualitative probability within T c , then ‘T unambiguous’requires that
the cardinal conditions in (7.8) and (7.9) obtain.
Observe that the above conditions do not involve u, which therefore has nothing to do
with ambiguity.
Next we relate Theorem 5.2 to the CEU model. Say that a capacity is convex-ranged
on A if for every A in A,
[0; A] = f B : B 2 A, B
Say that
Ag.
is continuous if for all events in S,
An & (\An ) if An & and
An % ([An ) if An % .
The following is largely a corollary of Theorem 5.2:
Corollary 7.3. Let be a CEU preference order with capacity satisfying (7.2) and subjectively unambiguous events A.
(a) Suppose that is continuous and convex-ranged on A. Then satis…es the axioms in
Theorem 5.2. Thus is probabilistically sophisticated on F ua relative to a convex-ranged
and countably additive probability measure p on A. Moreover,
= (p) on A for some
strictly increasing and onto map : [0; 1] ! [0; 1].
(b) Suppose in addition that (A0 ) = [0; 1] and that A0 is closed with respect to complements. Then = p on A and restricted to F ua has an expected utility representation.
(c) Suppose that is continuous and convex-ranged on A, that is exact and that
(A) + (Ac ) = 1 and 0 < (A) < 1 for some A 2 .
Then
= p on A and
A = A0 = A1 = A2 .
30
(7.10)
(7.11)
The key de…ning property of a qualitative probability on T c is ordinal additivity: For all events E, F
and G, subsets of T c such that G is disjoint from E and F , E
F if and only if (E [ G)
(F [ G).
21
Part (a) shows that the scope of Theorem 5.2 extends well beyond globally probabilistically
sophisticated preferences. Part (b) gives conditions under which the CEU preference order is
expected utility (and not merely probabilistically sophisticated) on the domain of unambiguous acts. Thus under the assumptions made here ambiguity is the only reason for deviating
from expected utility. However, only under the stronger conditions of part (c) can we conclude that A coincides with all classes of events discussed earlier. Di¤erences such as those
described in (7.5) are eliminated, in part, through the assumption that (A1 ) \ (0; 1) is nonempty.31 The equalities in (7.11) are valuable also because they provide explicit descriptions
of A under the stated assumptions.
Finally, consider risk and ambiguity attitudes for the CEU model. We make use of the
conjugate capacity , de…ned by
(E) = 1
(E c ) , for all E in
.
Corollary 7.4. Let
be a CEU order with utility index u and capacity satisfying the
conditions in part (a) of the previous corollary. Let p and be as provided there. Then:
(a) is ambiguity averse (loving) if and only if there exists a probability measure m on
satisfying
m( ) ( ) 1 ( ( )) on
and m( ) = p( ) on A;
(7.12)
is risk averse (loving) if and only if u is concave and (t)
t on [0; 1] (u is convex and
(t)
t on [0; 1]).
(b) A su¢ cient condition for ambiguity aversion (loving) is that, in addition to the maintained
assumptions, (its conjugate ) be exact and satisfy (7.10). In that case, is risk averse
(loving) if and only if u is concave (u is convex).
The corollary permits a comparison with Schmeidler’s (1989) de…nition of ambiguity and
risk aversion.32 In his approach, the capacity alone models the attitude towards ambiguity
and u alone models risk attitudes. For example, ambiguity aversion (as de…ned by Schmeidler) is equivalent to convexity of the capacity and risk aversion is equivalent to concavity of
u. In contrast, in our approach, part (a) of the corollary shows that the capacity, via , plays
a role in determining risk attitudes. Only in special cases, such as in part (b), are the separate roles claimed by Schmeidler for u and justi…ed. Moreover, while convexity of (which
implies exactness) is su¢ cient for ambiguity aversion given the additional assumptions in
(b), it is not necessary under those conditions.
One reason for this di¤erence in the two de…nitions is that Schmeidler’s is (formulated
and) motivated within the Anscombe-Aumann framework of two-stage, horse-race/roulettewheel acts, while intuition and interpretations that are appealing in this setting are not
easily transferred to the Savage domain. In this connection, Kreps (1988, p. 101) states
31
We owe this strong form of the result in (c) to Massimo Marinacci. Earlier versions of the paper assumed
(A1 ) = [0; 1] rather than merely (7.10).
32
Schmeidler does not de…ne ambiguity; thus a comparison with our de…nition of ambiguous event or act
is not pertinent.
22
that adoption of the Anscombe-Aumann domain is a problematic practice in descriptive
applications where only choices between Savage acts are observable. Epstein (1999) elaborates
on this point in the speci…c context of the meaning of ambiguity aversion. He shows, for
example, that convexity of the capacity is neither necessary nor su¢ cient for choice between
Savage acts that intuitively speaking re‡ects ambiguity aversion. In contrast, our approach
is focussed on behavior in the Savage domain and, we would argue, captures better intuition
related to that domain.
8. ALTERNATIVE DEFINITIONS
Some alternative approaches to de…ning ambiguity are examined here in light of the desiderata set out earlier and the limitations of our approach acknowledged in the introduction
(Section 1.4). In particular, connections to the literature on ambiguity are described.
8.1. Linearly Unambiguous
We motivated our de…nition in part by the suggestion that a necessary condition for an event
to be unambiguous is that it be ‘separable’ from events in its complement. The following
alternative de…nition embodies a stronger form of separability and therefore warrants some
attention:33
De…nition 8.1. An event T is linearly unambiguous if: (a) For all acts f 0 and f and all
outcomes z and z 0 ,
f 0 (s)
z
f 0 (s)
z0
if s 2 T c
if s 2 T
if s 2 T c
if s 2 T
f (s)
z
f (s)
z0
if s 2 T c
if s 2 T
if s 2 T c
if s 2 T
=)
(8.1)
;
and (b) The condition obtained if T is everywhere replaced by T c in (a) is also satis…ed.
Otherwise, say that T is linearly ambiguous.
It is apparent that if T is linearly unambiguous, then it is also unambiguous. The former
is a stronger property because the indicated invariance is required for all acts f 0 and f and
not just for the subclass of ‘conditional binary acts’as in (3.3). The economic signi…cance
of this di¤erence was touched upon in the discussion immediately following our de…nition
and is similar to the discussion in Section 4.2 of Machina and Schmeidler (1992). In any
event, it is apparent that linear ambiguity embodies a stronger form of separability than
does ambiguity. Why not use it as the key notion?
One answer is that (8.1) is too demanding to correspond to the intuitive notion of ambiguity. The invariance required by (8.1) may be violated because the decision-maker views
33
The de…nition recalls the statement of the Sure-Thing-Principle but di¤ers in that the ranking invariance
in (8.1) is required only for subacts z and z 0 that are constant on T .
23
outcomes in di¤erent states as complementary or substitutable for reasons that have nothing
to do with ambiguity. For example, she may be probabilistically sophisticated, thus assigning
probabilities to all events and translating any act into the induced lottery over outcomes,
but then she might evaluate the lottery by a risk-preference utility functional that is not
linear in probabilities (violating the Independence Axiom). Decision-makers who behave as
in the Allais Paradox are of this sort. Many events would be linearly ambiguous for such
decision-makers, though it seems intuitively that ambiguity has nothing to do with their
preferences. In contrast, for such (probabilistically sophisticated) decision-makers all events
are unambiguous (see the end of Section 3.2). Roughly speaking, our formal de…nition of
ambiguity relates to behavior in the Ellsberg Paradox but not the Allais Paradox, while linear ambiguity confounds the two.34 A similar point is made below regarding the de…nition
of ambiguity proposed by Ghirardato and Marinacci (1999).
That is not to dispute the potential usefulness of the notion of linear ambiguity. Zhang
(1997), who originated the notion, shows that it can help to provide an expected utility theory
that is ‘fully subjective’in the sense of the introduction. The linearity of the expected utility
function explains the choice of terminology.
8.2. ‘Equally Ambiguous’
Another alternative is motivated by the intuition that, if A and B are unambiguous, then it
should be the case that
A ` B () A [ C ` B [ C,
(8.2)
for all C disjoint from both A and B. It is easy to see that this equivalence is true given
our de…nition when C is also subjectively unambiguous, but it is not true more generally.
Therefore, one might employ (8.2) as the basis for a new de…nition. For example, say that
‘A and B are unambiguous’ if (8.2) is satis…ed for all suitably disjoint C. Evidently, this
de…nes a binary relation on events that might be better described in terms such as ‘A and
B are equally ambiguous’. Such a relation may prove interesting, but it evidently does not
deliver a notion of (absolute rather than relative) ambiguity; in particular, it does not deliver
a unique class of ‘unambiguous’events.35
There is another useful perspective on the above approach. A slight (and natural) variation on the above is to say that ‘A and B are equally ambiguous’ if for all x ; x; f and
g,
(x if A; x if B; f ( ) if (A [ B)c )
(x if A; x if B; f ( ) if (A [ B)c ) ()
(x if A; x if B; g( ) if (A [ B)c )
(x if A; x if B; g( ) if (A [ B)c ) .
34
Readers who attach little importance to the Allais Paradox may feel that such a view would justify
using (8.1) in place of (3.3). We feel that it argues for imposing a form of the Sure-Thing Principle on the
subdomain of unambiguous acts, rather than for changing the meaning of unambiguous.
35
The binary relation will typically not be an equivalence relation; for example, both ; and S are equally
ambiguous as any other event. We emphasize that we are not advocating this approach. However, something
along the lines of (8.2) may occur to some readers and thus we wish to clarify its relation to our de…nition.
24
(Here we are restricting attention to A and B that are mutually disjoint. Apart from this
restriction, one obtains (8.2) as the special case x
x, f ( ) = x, g( ) = x on C and
c
g( ) = x on (A [ B [ C) , where C is disjoint from A [ B.) In words, this equivalence
expresses the intuition that ambiguity has to do with “the e¤ect on own likelihood of a
change in the outcomes on other events”. Given that “own likelihood”refers to the ordinal
likelihood relation underlying preference, and we see no alternative possible interpretation,
then it is evident why this intuition relates at best a notion of relative ambiguity.
Another approach that leads to multiple collections of ‘unambiguous’events is the following:36 Though probabilistic sophistication may not prevail globally, because of ‘ambiguity’,
one might identify subdomains of acts where it does prevail and view each such subdomain
as consisting of unambiguous acts. This is essentially what is done by Sarin and Wakker
(1992) and Ja¤ray and Wakker (1994), with the exception that ‘expected utility’is used as
the reference model rather than ‘probabilistic sophistication’. Again, this approach at best
relates to ‘equally ambiguous’.37
Would it make sense to view all acts in the union of the above subdomains as unambiguous? We think not. Probabilistic sophistication on a subdomain of acts Fi re‡ects
exclusively the nature of the ranking of acts within Fi and similarly for any other subdomain Fj . On the other hand, the designation of Fi [ Fj as a set of unambiguous acts must
surely be based on some aspect of the ranking of acts in Fi relative to those in Fj (not to
mention also some acts that are not in Fj ). However, no such implications follow from probabilistic sophistication within each of Fi and Fj ; for example, probabilistic sophistication on
the union is not implied. We elaborate and illustrate this discussion in Section 9.2, where it
is also shown that de…ning unambiguous acts by a union as above leads to a counterintuitive
designation.
One could alternatively employ the intersection of such subdomains, or a suitable variant of the intersection, as a way to deliver a unique set of ‘unambiguous acts’. It would
occasionally be ‘far too small’, as in the example in Section 9.2. More importantly, the behavioral meaning of such a set seems totally unclear. In particular, such a de…nition, that is
expressed in terms of functional form representation, does not answer the question noted in
desideratum D1, namely “what behavior would indicate that the decision-maker perceives a
speci…c event or act as ambiguous?” Providing an answer to this question seems to us to be
the point. (This criticism applies equally to the union-based de…nition.)
36
See also the discussion following Theorem 5.2.
We emphasize that the two cited papers are not concerned explicitly with de…ning ambiguity and thus
our comments should not be interpreted as criticisms of these papers.
37
25
8.3. Ghirardato and Marinacci
Ghirardato and Marinacci (1999) de…ne ambiguity in terms of preference.38 They provide a
detailed comparison with our approach, but some comparison here seems appropriate.39
We begin with a de…nition of ambiguity and use it to de…ne attitudes towards ambiguity.
Ghirardato and Marinacci adopt the reverse order and de…ne ambiguity aversion (and loving)
…rst. One consequence of this reversal in order is that their de…nition of subjective ambiguity
applies only to preference orders that are either ambiguity averse or ambiguity loving, thus
excluding preferences where neither attitude prevails globally. At a conceptual level, we see
no justi…cation for such a limitation.
An indication of the di¤erence between the two de…nitions is that in the GhirardatoMarinacci approach, the absence of subjective ambiguity (A = ) is equivalent to preference
conforming to subjective expected utility theory, while for us it is equivalent to the probabilistic sophistication of preference.40 We try to identify the di¤erences in approach that
lead to this di¤erence in designations.
The essential problem in both papers is the distinction between risk and ambiguity.
Roughly speaking, we make the distinction through our behavioral de…nition of ambiguous
events and acts; all others are risky. Ghirardato and Marinacci make the distinction by
identifying the class of purely risky prospects or comparisons through their use of ‘cardinal
symmetry’. The essence of their approach is that the only rankings that clearly involve only
risk are those of the form
(x if A; x0 if Ac ) vs (y if A; y 0 if Ac ) ,
where x
x0 and y
y 0 , that is, comparisons involving 2 bets on the same event A but
having di¤erent stakes. Aspects of preference that are not tied down by such comparisons
are interpreted as involving a concern with ambiguity. As a consequence, they impute much
more importance to ambiguity as opposed to risk than do we.
To illustrate, consider a probabilistically sophisticated preference order that lies in both
the Choquet-expected-utility class and in the multiple-priors class; namely, take the special
Choquet utilities de…ned by (7.4) where is convex. According to their de…nition, all events
that are non-null and have non-null complements are necessarily ambiguous if preference
is not an expected utility order. To see the signi…cance of this designation, consider the
choice situation corresponding to the Allais Paradox. This situation is typically described
in terms of choice between lotteries. However, Savage (p. 103) translates it into an act
framework where objects of choice are acts over a state space. (Roughly, the decisionmaker is told there are 100 tickets in an urn, with numbers 1 through 100; and each act
promises a speci…ed prize depending on the number of the randomly drawn ticket. For
38
Ambiguity is taken as a primitive in Fishburn (1991) and Epstein (1999). Our focus is on models where
preference is the only primitive, corresponding to desideratum D1.
39
We focus exclusively on the de…nition of ambiguity. There are other major di¤erences; for example,
Ghirardato and Marinacci do not provide a counterpart of our Theorem 5.2.
40
We pointed out at the end of Section 3.2 that (global) probabilistic sophistication implies A = . The
converse is implied by Theorem 5.2.
26
example, one act pays $0 if the number is 1, $25 if the number lies between 2 and 11, and $5
otherwise.) If preference is probabilistically sophisticated as above and ‘nonlinear’in a way
that produces the paradoxical Allais-type choices, then the Ghirardato-Marinacci de…nition
would designate some events (subsets of ticket numbers) as ambiguous. Essentially, they
identify (nonindi¤erence to) ambiguity as the explanation for Allais-type choices.
Of course, it is possible that in the above choice situation, the decision-maker does not
accept the information provided about the content of the urn and that ambiguity is a factor
in her making the ‘paradoxical’ choices. It is very much an assumption on our part, that
is implicit in our de…nition, that ambiguity does not play a role in similar situations. Put
another way, an implicit assumption on our part is that a probabilistically sophisticated
agent who translates acts into induced lotteries over outcomes, evaluates those lotteries in
precisely the same way as she would evaluate those lotteries were they presented to her
as objective lotteries and hence as risky prospects. Though undoubtedly an assumption, a
similar identi…cation is common practice in the standard models whereby once a subjective
expected utility decision-maker has her prior, we typically view her as a vNM decision-maker
having the same vNM index. In any event, we are led to interpret an evaluation of induced
lotteries that is ‘nonlinear’ in the sense of deviating from vNM theory as re‡ective exclusively of preferences over risky prospects. From our perspective, the Ghirardato-Marinacci
approach confounds the Allais and Ellsberg Paradoxes, much as occurs with the notion of
linear ambiguity, and attributes ‘too much’importance to ambiguity as opposed to risk.41
9. ELLSBERG URNS
Further perspective on our approach to ambiguity is provided by examining choice situations
involving Ellsberg urns.
9.1. Two Colors
A single urn containing balls that are either red or blue is informative regarding our de…nition.
The decision-maker is told only the total number of balls in the urn and is to rank bets on
the color of a randomly drawn ball. The natural state space consists of the two points R and
B. If each is non-null, then both singleton sets are necessarily unambiguous according to the
formal de…nition.42 As for intuition, it is not clear. On the one hand, one might feel that
when no information is provided about the color composition of the urn, then many decisionmakers would perceive each singleton event as ambiguous. On the other hand, it is not clear
what behavior would re‡ect such ambiguity; and in the choice-theoretic tradition, it is only
such behavior that renders meaningful the designation of an event as ambiguous. Because we
have adopted the view that the behavioral manifestation of ambiguity is ‘nonseparability’,
it is not surprising that our de…nition leads to each singleton event being designated as
41
See Lemma D.1 in the appendix for a further illustration, within the concrete setting of CEU, of the
di¤erence between ‘ambiguity’according to the two de…nitions.
42
This presumes Savage’s axiom P 3.
27
unambiguous. Just as in consumer demand theory, if there are only two goods (and if
preference is suitably monotone), then each good is weakly separable, so too here there is
insu¢ cient scope for the separability required by our de…nition to have any bite.
9.2. Two Urns
This section describes an example based on Ellsberg’s 2-urn experiment. The example adds
further support for our de…nition by illustrating its tractability (the condition de…ning ‘subjectively unambiguous’may be checked in concrete settings) and by providing an intuitive
designation of ambiguous events.
However, its main purpose is the following: Theorem 5.2 shows that (given our axioms)
preference is probabilistically sophisticated on the set F ua of unambiguous acts. One might
wonder about the converse; that is, if is probabilistically sophisticated on F , the set of
all acts measurable with respect to some -system A , then are events in A unambiguous?
The example shows that the answer is no; in general, there may be several -systems A as
above with the class A of subjectively unambiguous events being only one of them.43 Thus,
speaking roughly, unambiguity is strictly stronger than (local) probabilistic sophistication.
A related conceptual point that is illustrated by the example is that whether or not an event
(or act) is deemed to be ambiguous depends on what is assumed observable by the analyst.
Consider the state space S = S1 S2 , where S1 = S2 =
and where the …nite set
represents the possible states in each urn Si , i = 1; 2. Let p be a probability measure on
with full support. The decision-maker is told that p describes the distribution of states
within the …rst urn S1 , but she is told less about the second urn S2 . For concreteness, take
outcomes X
R1 . Let Fi be the subset of F (the set of acts over S), consisting of acts
that are measurable with respect to i , the ith co-ordinate -algebra.44
In order to address the issues mentioned above, for example, in order to identify unambiguous events or subdomains where probabilistic sophistication prevails, we need to specify
a preference order
on F. In Appendix E, we specify
having the following features (if
45
j j 4):
1.
satis…es P 3; and the preference to bet on A1
43
A2 over B1
B2 is independent of
However, probabilistic sophistication on F implies that all events in A are unambiguous in the following restricted sense: The event T in A is A -unambiguous if it satis…es our de…nition when all acts
involved in the rankings (3.3) are restricted to be A -measurable. Moreover, A -unambiguity of all events
in A is also su¢ cient, given suitable reformulations of our axioms, for probabilistic sophistication on F .
This parallels the result that under suitable conditions, all events being unambiguous characterizes global
probabilistic sophistication (see Section 8.3). Such a characterization is a slight restatement of the MachinaSchmeidler characterization of global probabilistic sophistication. The local version outlined above has the
added unattractive feature of being self-referential (probabilistic sophistication on F is characterized by
means of properties relative to A ). In contrast, in our Theorem, A and probabilistic sophistication on F ua
are delivered in an explicit and constructive fashion.
44
S2 : B1 S1 g and similarly for 2 .
1 = fB1
45
If
consists of two elements, then all events are unambiguous for the reasons given in the previous
subsection. If consists of three elements, the properties below can be delivered under slightly strengthened
assumptions; for example, if the three elements have equal probability under p.
28
the stakes involved (Savage’s P 4), which implies a complete and transitive likelihood
relation ` on all such rectangles.
2. There is a strict preference for betting on any event E
(satisfying 0 < p(E) < 1)
when it is ‘drawn’from S1 rather than from S2 ; that is, E S2 ` S1 E.
3. The two urns are viewed as independent in the sense that (for all Ai ; A0i and Bi )
A1
A2
`
B1
A2 () A1
A02
`
B1
A02
A1
A2
`
A1
B2 () A01
A2
`
A01
B2 .
4.
is probabilistically sophisticated on F1 and every set in
5.
is probabilistically sophisticated on F2 and every set S1
0 < p(B2 ) < 1.
1
is unambiguous.
B2 in
2
is ambiguous if
The …rst three properties show that is intuitive, which justi…es interest in the lesson
to be drawn from the last two properties. The fourth illustrates the intuitive performance of
our de…nition of ambiguity. The …nal one shows that though preference is probabilistically
sophisticated on F2 , every ‘nontrivial’act in F2 (every ‘nontrivial’event in 2 ) is ambiguous.
There is clear intuition for the fact that probabilistic sophistication within F2 does not imply
unambiguity. Roughly, the noted structure re‡ects exclusively the decision-maker’s view of
events within 2 . On the other hand, whether or not events in 2 are subjectively ambiguous
depends also on how they are viewed relative to events outside 2 (such as the comparison
between drawing the event E from S2 as opposed to drawing it from S1 ).
Suppose, however, that the analyst can observe only the ranking of acts in F2 , which
can be identi…ed with acts over the state space S2 . Because preference is probabilistically
sophisticated on F2 , our de…nition applied to the restricted preference and state space would
imply that all events in S2 (equivalently, in 2 ) are unambiguous. This is perfectly natural in
a behavioral approach. Whether or not an event is revealed to the analyst to be ambiguous
depends on what is assumed observable by the analyst. In the present case, the subjective
ambiguity of S1 B2 in 2 is revealed only through the ranking of some acts not in F2 .
Finally, the example casts some light on alternative de…nitions of ambiguity. Recall the
discussion in Section 8.2 of the use of subdomains where probabilistic sophistication prevails
(here F1 and F2 ) as the basis for a de…nition of unambiguity. The union F1 [ F2 consists
of all acts that are based entirely on one urn, whether the …rst or the second. There is no
intuition for viewing all such acts as unambiguous. The intersection F1 \ F2 consists of
only constant acts. Thus using the intersection to de…ne ambiguity would result in subsets
of the …rst urn, where the probability law is told to the decision-maker, being designated
ambiguous. As for the Ghirardato-Marinacci de…nition, it agrees with ours in this example
in designating subsets of the second urn as ambiguous.
29
A. APPENDIX: -System
Proof of Lemma 5.1: The only nontrivial step is to show that if T1 and T2 are disjoint
unambiguous events, then T1 [ T2 is unambiguous. Suppose that for some disjoint subsets
A and B of (T1 [ T2 )c ; act h and outcomes x ; x; z; z 0 2 X , that
1
0
1
0
x if s 2 A
x if s 2 A
B x if s 2 B
C
B x if s 2 B
C
B
C
B
C
@ h if s 2 (T1 [ T2 )c n(A [ B) A
@ h if s 2 (T1 [ T2 )c n(A [ B) A :
z if s 2 T1 [ T2
z if s 2 T1 [ T2
Apply (3.3) in turn for T2 and then T1 to deduce
0
1
0
x if s 2 A
B x if s 2 B
C
B
B
C
B
c
@ h if s 2 (T1 [ T2 ) n(A [ B) A
@
0
z if s 2 T2 [ T1
x
x
h
z0
if
if
if
if
1
s2A
C
s2B
C.
c
s 2 (T1 [ T2 ) n(A [ B) A
s 2 T2 [ T1
Therefore, T2 [ T1 satis…es the appropriate form of (3.3).
Next prove that (3.3) is satis…ed also by (T1 [ T2 )c . By Small Unambiguous Event
Continuity (Axiom 4) applied to the unambiguous events T1 and T2 , there exists a partition
fAi gni=1 of S in A such that (T1 [ T2 )c equals the …nite disjoint union
[
(T1 [ T2 )c =
Ai :
Ai (T1 [T2 )c
Thus the …rst part of this proof establishes (3.3) for (T1 [ T2 )c .
To complete the proof, it su¢ ces to show that for any fAn g1
n=1 , a decreasing sequence in
1
x: Then
A, we have \i=1 A 2 A: By Nondegeneracy, there exist two outcomes x
x
x
fn =
if s 2 An
if s 2 Acn
2 F ua , for all n = 1; 2; :::
By Monotone Continuity (Axiom 5),
f1 =
x
x
if s 2 \1
n=1 An
c
if s 2 (\1
n=1 An )
2 F ua .
Consequently, \1
n=1 An 2 A:
B. APPENDIX: Existence of Probability
The …rst step in proving Theorem 5.2 is to prove the existence of a probability measure
representing ` . This appendix states a theorem (proven by Zhang (1999)) that delivers
such a probability measure given suitable properties for ` . The theorem extends Theorem
30
14.2 of Fishburn (1970) to the present case of a -system of events. Next it is shown that
these properties are implied by the axioms adopted for , as speci…ed in Theorem 5.2.
For the following theorem, A denotes any -system and ` is any binary relation on A;
that is, they are not necessarily derived from , though the subsequent application is to that
case. De…ne N (;) = fA 2 A : A ` ;g:
Theorem B.1. There is a unique …nitely additive, convex-ranged probability measure p on
A such that
A ` B () p(A) p(B); 8 A; B 2 A
if (and only if)
`
satis…es the following:
F1 : ;
`
A; for any A 2 A.
F2 : ;
`
S.
F3 :
`
is a weak order.
F 4 : If A; B; C 2 A and A \ C = B \ C = ;; then A
`
B () A [ C
`
B [ C.
F4’ : For any two uniform partitions fAi gni=1 and fBi gni=1 of S in A, [i2I Ai
jIj = jJj:
`
[i2J Bi , if
F 5 : (i) If A 2 AnN (;), then there is a …nite partition fA1 ; A2 ; :::; An g of S in A such
that (1) Ai A or Ai Ac ; i = 1; 2; :::; n; (2) Ai ` A; i = 1; 2; :::; n:
(ii) If A; B; C 2 AnN (;) and A \ C = ;; A ` B; then there is a …nite partition
fC1 ; C2 ; :::; Cm g of C in A such that A [ Ci ` B; i = 1; 2; :::; m:
F6 : If fAn g is a decreasing sequence in A and if A
in A, then there exists N such that A ` An
\1
` A for some A and A
0 An
A for all n N .
`
`
Axioms F 1, F 2, F 3 and F 4 are similar to those in Fishburn’s Theorem 14.2, while F 5
strengthens the corresponding axiom there. The additional axioms F 40 and F 6 are adopted
here to compensate for the fact that A is not a -algebra.
For the remainder of the appendix, A, ` and are as speci…ed in Theorem 5.2 and the
axioms stated in (a) are assumed. By Lemma 5.1, A is a -system. The objective now is
to prove that conditions F 1 F 6 are implied by the axioms given for . Proofs that are
elementary are not provided.
Lemma B.2. Let fAi g be a uniform partition of S in A. Then for all outcomes fxi g and
for all permutations ;
x (i) ; Ai i (xi ; Ai )i .
(B.1)
In other words, every uniform partition is strongly uniform.
31
Proof. Without loss of generality, assume x1
0
1
x1 if s 2 A1
B x2 if s 2 A2 C
B
C
B x3 if s 2 A3 C
B
C
@ ... ...
A
xn if s 2 An
x2 and that
0
x2 if s 2 A1
B x1 if s 2 A2
B
B x3 if s 2 A3
B
@ ... ...
xn if s 2 An
1
C
C
C.
C
A
Since fAi gni=3 are unambiguous, the appropriate form of (3.3) implies
x1
x2
that is, A1
`
if s 2 A1
if s 2 Ac1
x2
x1
if s 2 Ac2
if s 2 A2
;
A2 , a contradiction. Similarly for the other cases.
By showing that the axioms 2-6 for
imply properties F 1
F 6 for
`,
we prove:
Theorem B.3. Let be a preference order on F and denote by A the set of all unambiguous
events. If
satis…es Axioms 2-6, then there exists a unique convex-ranged and countably
additive probability measure on A such that
A
`
B
()
p(A)
p(B), for all A; B 2 A.
Proof. Fix outcomes x
x. Properties F 1 - F 3 for ` are immediate.
F4: (Note the role played here by the speci…c de…nition of A; not any -system would do.)
If A ` B, then
x if s 2 A
x if s 2 B
, or
x if s 2 Ac
x if s 2 B c
1
1 0
0
x if s 2 AnB
x if s 2 AnB
C
C B x if s 2 BnA
B x if s 2 BnA
C
C B
B
@ h if s 2 (A \ B) [ (C c n(A [ B)) A @ h if s 2 (A \ B) [ (C c n(A [ B)) A ;
x if s 2 C
x if s 2 C
where h = (x if A \ B; x if C c n(A [ B)). Since
0
1 0
x if s 2 AnB
B x if s 2 BnA
C B
B
C B
@ h if s 2 (A \ B) [ (C c n(A [ B)) A @
x if s 2 C
C is unambiguous,
x
x
h
x
if
if
if
if
1
s 2 BnA
C
s 2 AnB
C,
s 2 (A \ B) [ (C c n(A [ B)) A
s2C
or A [ C ` B [ C. Reverse the argument to prove the reverse implication.
F40 follows from Lemma B.2 and Axiom 6.
F5 (i): Since A ` ;;
x
x
if s 2 A
if s 2 Ac
x=
32
x if s 2 A
x if s 2 Ac
:
By Small Unambiguous Event Continuity (Axiom 4), there is a partition fAi gni=1 of S in A,
re…ning fA; Ac g and such that
if s 2 A
if s 2 Ac
x
x
if s 2 Ai
if s 2 Aci
x
x
, i = 1; 2; :::; n.
That is, for each i, Ai A or Ai Ac and in addition, Ai ` A.
F5 (ii): Let A, B and C be in AnN (;), A \ C = ; and A ` B. Then
0
1
x if s 2 A
x if s 2 B
x if s 2 A
A = g:
f=
= @ x if s 2 C
x if s 2 B c
x if s 2 Ac
c
x if s 2 A nC
By Small Unambiguous Event Continuity, there exists a partition fC1 ; :::; Cn g of S in A,
re…ning fA; C; Ac nCg and such that
x
x
f=
If Ci
if s 2 B
if s 2 B c
if s 2 Ci
if s 2 Cic
x
g
; i = 1; 2; :::; n:
C, then
x
x
0
x
@ x
x
if s 2 B
if s 2 B c
x
g
1
if s 2 Ci
A=
if s 2 A
c
if s 2 (A [ Ci )
if s 2 Ci
if s 2 Cic
x
x
=
if s 2 A [ Ci
if s 2 (A [ Ci )c
,
implying that A [ Ci ` B.
F6: Implied by Monotone Continuity.
C. APPENDIX: Proof of Main Result
Necessity of the axioms in Theorem 5.2: The necessity of Nondegeneracy and Weak
Comparative Probability is routine. Denote by D the order on Dpua (X ) represented by W .
Small Unambiguous Event Continuity: Let f
g and x be as in the statement of
the axiom. Denote by P = (x1 ; p1 ; :::; xn ; pn ) and Q the probability distributions over
outcomes induced by f and g respectively, and let x be a least preferred outcome in
fxg [ fx1 ; x2 ;...; xn g. Since P D Q D x , mixture continuity and monotonicity with
respect to stochastic dominance ensure there exists some su¢ ciently large integer N such
that W (1 N1 )P + N1 x > W (Q): Because p is convex-ranged, we can partition each set
N
Ai into N equally probable events fAij gN
j=1 in A. Let Ck = [i=1 Aik for k = 1; 2; :::; N .
Then fCk gN
1=N )pi for
k=1 is a partition of S in A, p(Ck ) = 1=N and p(Ai nCk ) = (1
each i and k. Consequently, [x if s 2 Ck ; f if s 2
= Ck ] induces the probability distribution
33
(1=N ) x + (1 1=N )P which is strictly preferred to Q. Combined with monotonicity with
respect to …rst-order stochastic dominance, this yields [x if s 2 Ck ; f if s 2
= Ck ]
[x if
s 2 Ck ; f if s 2
= Ck ] g: Similarly for the other part of the axiom.
1
Monotone Continuity: Given a decreasing sequence fAn g1
n=1 in A, p(An ) & p(\1 Ai ) by
the countable additivity of p. The required convergence in preference is implied by mixture
continuity of W . The limit f1 lies in F ua because we are given that A is a -system.
Strong-Partition Neutrality: Immediate from (5.1).
Su¢ ciency of the axioms in Theorem 5.2: Let p be the measure in Theorem B.3.
Lemma C.1. For unambiguous events A and B:
(a) A is null i¤ A ` ;.
(b) If A ` B ` ; and A \ B = ;; then A [ B
Proof. (a) Fix x
;.
y. Let A be null. Then
x if s 2 A
y if s 2 Ac
implying that A
`
`
y if s 2 A
y if s 2 Ac
x if s 2 ;
y if s 2 S
=y=
;
;. If A is not null, then by P 3,
x if s 2 A
y if s 2 Ac
implying that A ` ;.
(b) Let x y and A [ B
`
y if s 2 A
y if s 2 Ac
;, that
0
x
@ x
y
By (a), A and B are null and
0
1
y A
A
y=@ y B
c
y (A [ B)
This is a contradiction.
1
A
A
B
c
(A [ B)
x
@ y
y
Pf = x1 ; p(f
,
is,
0
For each f 2 F ua , de…ne
x if s 2 ;
y if s 2 S
=
1
1
A
A
B
c
(A [ B)
y:
0
x
@ x
y
(x1 )); :::; xn ; p(f
1
1
A
A
B
c
(A [ B)
(xn )) .
Because p is …xed, it may be suppressed in the notation. Accordingly, write
Pf 2 Dua (X ) = fPf : f 2 F ua g.
De…ne the binary relation
P
D
D
on Dua (X ) by
Q if 9 f
g, P = Pf and Q = Pg .
34
y:
Lemma C.2. If Pf = Pg ; then f
g. Thus
D
is complete and transitive.
Proof. We must prove that for any two partitions fAi gni=1 and fBi gni=1 of S in A, if Ai ` Bi ,
i = 1; 2; :::; n, then for all outcomes fxi gni=1 ,
0
1 0
1
x1 if s 2 A1
x1 if s 2 B1
B x2 if s 2 A2 C B x2 if s 2 B2 C
B
C B
C.
(C.1)
@ ... ...
A @ ... ...
A
xn if s 2 An
xn if s 2 Bn
Case 1: fAi gni=1 and fBi gni=1 are uniform partitions of S in A. The desired conclusion
follows from Lemma B.2 and Axiom 6.
Case 2: All probabilities fp(Ai )gni=1 and fp(Bi )gni=1 are rational. Because p is convex-ranged,
m
there exist fEj gm
j=1 and fCj gj=1 , two uniform partitions of S in A, such that
Ai =
[
Ej , i = 1; 2; :::; n and
Ej Ai
Bi =
[
Cj , i = 1; 2; :::; n.
Cj B i
Now Case 1 may be applied.
Case 3: This is the general case where some of the
irrational. Suppose contrary to (C.1) that
0
1 0
x1 if s 2 A1
x1
B x2 if s 2 A2 C B x2
C B
f =B
@ ... ...
A @ ...
xn if s 2 An
xn
probabilities p(Ai ) or p(Bi ) may be
1
if s 2 B1
if s 2 B2 C
C = g.
A
...
if s 2 Bn
(C.2)
Without loss of generality, assume that xn
x2 x1 and p(A1 ) = p(B1 ) is irrational.
By the convex range of p over A, there are rational numbers rm %m and two increasing
m 1
m
1
A1 and B1m B1 ; m = 1; 2; :::, such that
sequences fAm
1 gm=1 and fB1 gm=1 in A with A1
m
m
p(A1 ) = p(B1 ) = rm % p(A1 ) = p(B1 ) as m ! 1. Accordingly,
m
p(A1 nAm
1 ) = p(B1 nB1 ) = p(A1 )
p(Am
1 ) & 0 as m ! 1:
1
m 1
Thus, both fA1 nAm
1 gm=1 and fB1 nB1 gm=1 are decreasing sequences in A and
m
\1
m=1 (A1 nA1 )
`
m
\1
m=1 (B1 nB1 )
`
;:
De…ne
0
x1
@
x2
gm =
g
1
0
x1
if s 2 B1m
m A
@
x2
if s 2 B1 nB1
, g1 =
g
if s 2 B1c
35
1
m
if s 2 \1
m=1 B1
m A
if s 2 B1 n(\1
.
m=1 B1 )
if s 2 B1c
(C.3)
By Lemma C.1 and (C.3),
g1
g
f.
By Monotone Continuity, gm converges to g1 in preference as m
there exists an integer N1 such that
gm
In particular,
gN1
By P 3,
0
x1
@ x2
f
Therefore,
1
1
if s 2 AN
1
1 A
if s 2 A1 nAN
1
c
if s 2 A1
0
x1
@ x2
f
0
f whenever m
x1
@
=
x2
g
0
x1
@ x1
f
N1 :
1
if s 2 B1N1
if s 2 B1 nB1N1 A
if s 2 B1c
1
1
if s 2 AN
1
1 A
=f
if s 2 A1 nAN
1
c
if s 2 A1
1
1
if s 2 AN
1
1
A
if s 2 A2 [ (A1 nAN
1 )
c
if s 2 (A1 [ A2 )
0
x1
@ x2
g
! 1. Conclude that
f:
0
x1
@ x2
g
1
if s 2 B1N1
if s 2 B1 nB1N1 A :
if s 2 B1c
1
if s 2 B1N1
if s 2 B2 [ (B1 nB1N1 ) A .
if s 2 (B1 [ B2 )c
N1
N1
1
Note further that A2 [ (A1 nAN
` B2 [ (B1 nB1 ) since p(A2 [ (A1 nA1 )) = p(B2 [
1 )
(B1 nB1N1 )). Thus a proof by induction establishes that
1
1 0
0
1
x1 if s 2 B1N1
x1 if s 2 AN
1
B x2 if s 2 AN2 C B x2 if s 2 B N2 C
2
2
C
C B
B
B x3 if s 2 AN3 C B x3 if s 2 B N3 C ;
3
3
C
C B
B
A
A @ ... ...
@ ... ...
Nn
Nn
xn if s 2 Bn
xn if s 2 An
where AiNi
2.
`
Ni
i
BiNi , i = 1; 2; :::; n and every p(AN
i ) = p(Bi ) is rational, contradicting Case
The rest of the proof is similar to Steps 2-6 in the proof of Machina and Schmeidler’s
Theorem 2. For example, in the proof of mixture continuity of D on Dua (X ) (Step 3), Small
Unambiguous Event Continuity may be used in place of Savage’s P 6 in order to overcome
the lack of a -algebra structure for A.
36
D. APPENDIX: Choquet Expected Utility
Use ‘+’to denote disjoint union.
Proof of Lemma 7.2: Su¢ ciency may be proven by a tedious but routine veri…cation.
We prove necessity. It is convenient to express the conditions in the following equivalent
form: For all pairwise disjoint events A, B, C and D, each disjoint from T ,
(A [ D)
(B [ D) () (A [ D [ T )
if ( (A [ D)
(B [ D)) ( (A [ D [ C)
(A [ D)
(A [ D [ C)
(B [ D) =
(A [ D [ T )
(B [ D [ T ); and
(B [ D [ C)) < 0, then
(B [ D [ C) = (A [ D [ C [ T )
and the above conditions are satis…ed also by T c .
Let T be unambiguous. If x
x, then (A + D)
x
x
0
x
B x
B
@ x
x
x
x
A+D
Sn(A + D)
x
x
1
A+D
C
B
C
c
T n(A + B + D) A
T
B+D
Sn(B + D)
0
x
B x
B
@ x
x
A+D+T
Sn(A + D + T )
(A + D + T )
x
x
(B [ D [ T ) and
(B [ D [ C [ T ) ;
(D.1)
(D.2)
(D.3)
(D.4)
(B + D) i¤
()
1
B+D
C
A
C ()
c
T n(A + B + D) A
T
B+D+T
Sn(B + D + T )
()
(B + D + T ).
Suppose next that (D.2) is satis…ed. In fact, suppose that
(A + D) < (B + D) and (A + D + C) > (B + D + C).
(The other case is similar.)
The event T is unambiguous only if
0
1
x
if s 2 A
B x
C
if s 2 B
B
C
B y
C
if s 2 C
B
C
@ h(s) if s 2 T c n(A + B + C) A
z
if s 2 T
0
B
B
B
B
@
x
x
y
h(s)
z
if s 2 A
if s 2 B
if s 2 C
if s 2 T c n(A + B + C)
if s 2 T
1
C
C
C ()
C
A
(D.5)
(D.6)
a similar ranking obtains for the acts where z 0 replaces z. Suppose that h equals y on D
and y on T c n(A + B + C + D) and that
y
x
y
x
y, z = y and z 0 = x .
37
(D.7)
Then by the de…nition of Choquet integration, the above equivalence becomes
u(x ) (A + D) + u(y) [ (C + T + A + D)
u(x) [ (B + C + T + A + D)
(C + T + A + D)]
u(x ) (B + D) + u(y) [ (C + T + B + D)
u(x) [ (B + C + T + A + D)
(A + D)] +
(B + D)] +
(C + T + B + D)]
if and only if
u(x )
u(x) [
u(x )
u(x) [
(T + A + D) + u(y) [ (C + T + A + D)
(T + A + D)] +
(B + C + T + A + D)
(C + T + A + D)]
(T + B + D) + u(y) [ (C + T + B + D)
(T + B + D)] +
(B + C + T + A + D)
(C + T + B + D)].
Equivalently,
[u(x ) u(y)] ( (A + D)
(B + D)) +
[u(y) u(x)] ( (C + T + A + D)
(C + T + B + D))
0
[u(x ) u(y)] ( (T + A + D)
(T + B + D)) +
[u(y) u(x)] ( (C + T + A + D)
(C + T + B + D))
0,
i¤
where this equivalence obtains for all outcomes.
By (D.5) and appropriate forms of (D.1), which has already been proven, conclude that
( (A + D)
(B + D)) and ( (T + A + D)
(T + B + D)) are both negative, while
( (C + A + D)
(C + B + D)) and ( (C + T + A + D)
(C + T + B + D)) are both
positive. Because the range of u has nonempty interior, we can vary the above utility values
su¢ ciently to conclude from the preceding equivalence that
(T + A + D)
(T + B + D) = (A + D)
(D.8)
(B + D).
Next apply a similar argument for the case
y
x
y
x
y, z = x and z 0 = x ,
in place of (D.7). One obtains the equivalence
[u(x ) u(y)] ( (A + D)
(B + D)) +
[u(y) u(x)] ( (C + A + D)
(C + B + D))
0
i¤
[u(x ) u(y)] ( (T + A + D)
(T + B + D)) +
[u(y) u(x)] ( (C + T + A + D)
(C + T + B + D))
38
0,
where this equivalence obtains for all outcomes. Apply (D.8) and conclude that
(C + A + D)
(C + B + D) = (C + T + A + D)
(C + T + B + D).
The following lemma refers to alternative de…nitions of ambiguity described in Section 8.
Its proof is immediate given Lemma 7.1 and results from Zhang (1997) and Ghirardato and
Marinacci (1999).
Lemma D.1. (a) If A0 is closed with respect to complements, then A0 coincides with the
class of linearly unambiguous events and A0
A.
(b) If is exact, then A1 coincides with the class of events unambiguous in the sense of
either Ghirardato-Marinacci or linear ambiguity and A1 = A0
A.
Proof of Corollary 7.3: (a) The assumptions on imply that
satis…es the axioms in
Theorem 5.2. (Continuity implies Monotone Continuity for
and convex-ranged implies
Small Unambiguous Event Continuity.) Therefore, there exists a convex-ranged and countably additive p representing the likelihood relation on A that is implicit in . Conclude that
p must be ordinally equivalent to on A.
(b) From (a) and Lemma D.1,
= (p) on A
A0 , where p is convex-ranged on A.
Therefore, (A0 ) = (p(A0 )) = [0; 1]. Because is (strictly) increasing and onto, conclude
that p(A0 ) = [0; 1]. Now it is straightforward to prove that is the identity function. (For
any two x1 ; x2 2 [0; 1] with x1 +x2 1, there exist A1 2 A0 and A2 2 A such that p(A1 ) = x1 ,
p(A2 ) = x2 and A1 \ A2 = ;: From the de…nition of A0 , (A1 [ A2 ) = (A1 ) + (A2 ), or
(A1 [ A2 ) =
=
=
=
(p(A1 [ A2 )) = (p(A1 ) + p(A2 ))
(x1 + x2 ) = (A1 ) + (A2 )
(p(A1 )) + (p(A2 ))
(x1 ) + (x2 ):
Since is continuous, is linear on [0; 1].)
(c) Let p be the measure provided by (a) and …x any measure q in core( ). Continuity of
implies that q is countably additive (Schmeidler (1972)). By Lemma D.1, the event A
satisfying (7.10) lies in A. We show that it satis…es also: For any B 2 A,
p(B) = p(A) =) q(B) = q(A).
(D.9)
Then a recent result by Marinacci (1998) allows us to conclude that p = q. Because this
is true for any q in the core, conclude further that core( ) = fpg and hence, because is
exact, that
= p on A.
From above, A
A2 , the class of events where all measures in the core agree. In addition,
A1
A by Lemma D.1 and A2 = A1 = A0 by Lemma 7.1.
39
Thus it su¢ ces to prove (D.9). Let p(B) = p(A). Then
(B) =
(D.10)
(p(B)) = (p(A)) = (A).
Similarly, (B c ) = (Ac ). From the hypothesis (A) + (Ac ) = 1, deduce that
(B) + (B c ) = 1.
Because q( )
( ), deduce further that q(B) =
q(B) = q(A) from (D.10).
(B). Similarly, q(A) =
(A). Finally,
Proof of Corollary 7.4: (a) The proof regarding ambiguity aversion is similar to the
proof of Epstein (1999, Lemma 3.4). For su¢ ciency of (7.12), take the probabilistically
sophisticated order ps required by (6.2) to be the counterpart of (7.4), that is, the CEU
function with utility index u and capacity
(m). For risk attitudes, note that on F ua , is
R
represented by the utility function u(h)d (p), which is the rank-dependent functional form
familiar from the literature on ‘non-expected utility’ theories of preference over lotteries.
Therefore, the asserted characterization of attitudes towards risk follows from Chew, karni
and Safra (1987), for example.
(b) Prove ambiguity aversion; the other case is similar. Because A = A2 , p can be extended
to a measure m on such that m 2 core( ). For the probabilistically sophisticated order
ps
required by (6.2), take the expected utility order with probability measure m and vNM
index u. Then it su¢ ces to show that
Z
Z
u(f ) dm
u(f ) d
for all acts f in F. This is true because
n 1
i=1 [ u(xi )
R
u(f ) d
u(xi+1 ) ] m([ij=1 Ej )
R
u(f ) dm =
([ij=1 Ej )
0,
where f = xi on Ei , i = 1; :::; n and u(x1 ) > :::: > u(xn ) and where the non-negativity is
due to the fact that m lies in the core of .
E. APPENDIX: A Particular Preference Order
We de…ne a preference order satisfying the properties described in Section 9.2. As a …rst
step in de…ning utility over F, let U2 : F2 ! R1 be de…ned by
Z
U2 (f ) =
u(f ) d (p), f 2 F2 ,
(E.1)
S2
where u is a continuous and strictly increasing vNM index, where : [0; 1] ! [0; 1] is
a strictly increasing and onto map and integration is in the sense of Choquet (Section 7).
Assume further that (t) < t for all t 2 (0; 1).
40
To de…ne U on F, observe that given any act f over S and s1 2 S1 , the restriction f (s1 ; )
can be viewed as an act over S2 , giving meaning to U2 (f (s1 ; )). Thus U can be de…ned as
follows: For each f 2 F,
Z
Z Z
U (f ) =
U2 (f (s1 ; )) dp(s1 ) =
u(f ) d (p(s2 )) dp(s1 ).
(E.2)
S1
S1
S2
Let
be the preference order represented by . The asserted properties can be veri…ed.
Of particular interest is the assertion regarding the ambiguity of events in 2 . Its proof
is straightforward, though tedious (it is available upon request from the authors). To show
that a given T is ambiguous, one must provide disjoint events A and B, each disjoint from
T , an act h and outcomes x , x, z and z 0 such that the invariance in (3.3) is violated when
z is replaced by z 0 . Figure 1 illustrates schematically the kinds of events that work for the
T shown there.46 To see intuitively why the noted invariance is violated, take the act h to
be constant and equal to y on T c n(A [ B) (this su¢ ces for the proof), suppose that B1 is
empty and let x
y
x. If z = x , then the best outcome x is attained on an event
containing [0; t1 ] S2 , and the latter has objective probability p ([0; t1 ]). This precision may
lead to the preference for the conditional bet on A rather than on B (equal here to B2 ) and
hence to the …rst ranking shown in (3.3). However, if z is replaced by z 0 = y, the above
perspective is changed and a reversal in ranking may occur.
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In the Figure, each urn is taken to be the unit interval and B = B1 [ B2 and C = T c n(A [ B).
41
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42